of \(L^k\) . The socle of \(L_k\)  is a direct product of \(k\)  minimal normal
subgroups of \(L_k\) , each of them isomorphic to \(N\) , and that
\(L_k/N^k\cong L/N\) . The quotient of \(L_{k+1}\)  over a minimal normal
subgroup of \(L_{k+1}\)  is isomorphic to \(L_k\) .
We have that \(\operatorname{d}(L_1)\le \operatorname{d}(L_2)\le \operatorname{d}(L_3)\le \cdots \)  and, according to [1]},
\(\operatorname{d}(L_{k+1})\le \operatorname{d}(L_k)+1\)  for every \(k\) . Hence, if \(d\ge \operatorname{d}(L)\) , then
there is a unique \(k\)  such that \(\operatorname{d}(L_k)=d<\operatorname{d}(L_{k+1})\) . Let us call
\(f(d)=k+1\) . Then, by [2]},
\(f(d)=1+\log _q\bigl (\vert N\vert ^{d-1}\big /\vert \operatorname{H}^1(L/N,N)\vert \bigr ),\) 
